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Gow-Tamburini type generation of the special linear group for some special rings. | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 13، شماره 2، شهریور 2024، صفحه 123-132 اصل مقاله (417.89 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2023.134366.1800 | ||
| نویسندگان | ||
| Naresh Vasant Afre1؛ Anuradha S. Garge* 2 | ||
| 1Department of Mathematics, University of Mumbai, Mumbai, India | ||
| 2Department of Mathematics, University Mumbai, Kalina Campus, Mumbai, India | ||
| چکیده | ||
| Let $R$ be a commutative ring with unity and let $n\geq 3$ be an integer. Let $SL_n(R)$ and $E_n(R)$ denote respectively the special linear group and elementary subgroup of the general linear group $GL_n(R).$ A result of Hurwitz says that the special linear group of size atleast three over the ring of integers of an algebraic number field is finitely generated. A celebrated theorem in group theory states that finite simple groups are two-generated. Since the special linear group of size atleast three over the ring of integers is not a finite simple group, we expect that it has more than two generators. In the special case, where $R$ is the ring of integers of an algebraic number field which is not totally imaginary, we provide for $E_n(R)$ (and hence $SL_n(R)$) a set of Gow-Tamburini matrix generators, depending on the minimal number of generators of $R$ as a $Z$-module. | ||
| کلیدواژهها | ||
| Quadratic extensions؛ ring of integers of number fields؛ Special linear group؛ Elementary subgroup | ||
| مراجع | ||
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[1]T. Y. Lam, Serre’s problem on projective modules, Springer Monographs in Mathematics, Springer-Verlag, Berlin, | ||
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